I would recommend against it, as github was specifically setup to work with git (hence its name). Git has been set up to handle pretty complex projects, while butler is only a quick update program and set up to handle full releases, where git is for handling active projects and on which many people work too. So it ain't so strange that getting the full grasp how how git works is harder than to understand butler.
And the only good way for butler to support github will in the end still be to use git as a dependency and even if it's possible to use butler for the whole update you would when connection issues is why you want it get the same issues as butler would be bound to the same protocol and connection methods git uses, this because this is simply how the git network is set up, and all clients making use of that are therefore bound by the same rules. So Butler will not be able to solve your connection issues, unfortunately.
Git is after all a version control system, and its network is setup the way it is for a reason. I therefore deem it likely that if you go to other git-based networks (like GitLabs or BitBucket) you will likely have the same issues.
I think it will be more fruitful to find out why you have these connection issues with git and try out if git itself also provides ways to deal with those. Stability issues with git are not something that should happen, since most package managers use the git network to keep their data in how to install packages and if versions are still up to date etc. Unix based systems therefore rely heavily on this, and there are also such managers for Windows (Scoop and Chocolatey) and as far as I know they are also reliant on git. I deem it also possible you should have a chat with your ISP.
Trying to convert a tool to do that for which it was not intended to do, will mostly only undermine its primary function. And since Butler is extremely robust for what it was set out to do (at least as far as I experienced so far), I don't think we should undermine that, especially not since the tools that were set up for the job are likely not where the "evil" lies.